This PDF covers the following topics related to Groups, Rings and Fields
: Familiar algebraic systems: review and a look ahead, Binary operations, and a
first look at groups, Interlude: properties of the natural numbers, Integers,
Polynomials, Equivalence relations, and modular arithmetic.
This PDF covers the following topics related to Rings and
Fields : A pseudo-historical note, More on subrings and ring
homomorphisms, The evaluation or the substitution map, Defining fractions, Using
the universal property of the field of fractions, An application of the first
isomorphism theorem, The factor theorem and the generalized factor theorems,
Gaussian integers, Irreducibility and zeros of polynomials,
Content of a polynomial with rational coefficients, An example on the mod
irreducibility criterion, Factorization: uniqueness, and prime elements, Ring of
integer polynomials is a UFD, Greatest common divisor for UFDs, Extension of
isomorphisms to splitting fields, Finite fields, etc.
Author(s): Alireza Salehi
Golsefidy, University of California San Diego
On the one hand this
book intends to provide an introduction to module theory and the related
part of ring theory. Topics covered includes: Elementary properties of
rings, Module categories, Modules characterized by the Hom-functor, Notions
derived from simple modules, Finiteness conditions in modules, Dual
finiteness conditions, Pure sequences and derived notions, Relations between
functors and Functor rings.
This wikibook explains ring theory. Topics
covered includes: Rings, Properties of rings, Integral domains and Fields,
Subrings, Idempotent and Nilpotent elements, Characteristic of a ring,
Ideals in a ring, Simple ring, Homomorphisms, Principal Ideal Domains,
Euclidean domains, Polynomial rings, Unique Factorization domain, Extension
this book is to help the students by giving them some exercises and get them
familiar with some solutions. Some of the solutions here are very short and in
the form of a hint. Topics covered includes: Sets, Integers, Functions, Groups,
Rings and Fields.
note covers the following topics: Introduction to number rings, Ideal
arithmetic, Explicit ideal factorization, Linear algebra for number rings,
Geometry of numbers, Zeta functions, Computing units and class groups, Galois
theory for number fields.